I never understood why you are "black" if only your father or mother was white. It's a notion that seems to be unique to black America, I haven't encountered any other examples where one race/culture socially eclipses the other completely like that.

If would be like if the children of a South African and an Italian were told that they must be only Italian, just because.

It's a caste enforcement technique to keep the poor blacks in a lower status than poor whites, so they wouldn't unite against their common oppressor. Not unique to US, but maybe more persistent.
French slave owners came up with racial divisions of octaroon, quadroon, mulatto, etc.
Spanish and Portuguese South America also kept pretty close tabs on racial purity, especially during the times of liberation, emancipation, and decolonization.
Native/aboriginal people have also been racially subjugated everywhere under colonization/empire systems. Australia, for instance.

to keep the poor blacks in a lower status than poor whites

Important adjustment to an otherwise very good assessment.

It probably follows from patrilineality, the idea that lineage derives from the bloodline of the males. It affects inheritance of so many things. No idea where that tradition comes from though

Thank you! That's exactly what I was looking for and I learned a new word too:

"and is an example of _hypodescent_, the automatic assignment of children of a mixed union between different socioeconomic or ethnic groups to the group with the lower status."